A family of weightwise (almost) perfectly balanced boolean functions with optimal algebraic immunity

The main cryptographic features of Boolean functions when the input is restricted to some subset of Fn2 are studied recently because of the innovative stream cipher FLIP M ' eaux et al. (2016). In this paper, we propose a large family of Boolean functions which are (almost) balanced on every set of vectors in Fn2 \ {0, 1} with constant Hamming weight (the so-called weightwise (almost) perfectly balanced, W(A)PB). We show that these W(A)PB functions have optimal algebraic immunity on Fn2 and good algebraic immunity on some subsets of vectors in Fn2, especially on the subsets of vectors with constant Hamming weight. This is the first time that W(A)PB functions with good local algebraic immunities are presented. Moreover, we discuss the nonlinearity and weightwise nonlinearity of these functions.